In this subsection we study another interesting limit called intermediate temperature limit which is defined by \(l^{\prime }T \ll 1\ll lT \)  or equivalently \(r_H\ll r^*_{l^{\prime }}\)  and \(r^*_{l^{\prime }+2l}\rightarrow r_H\) . We consider this limit at high energy \(l\Lambda _c\ \&\ l^{\prime }\Lambda _c\ll 1\) . Using eqs. (REF ), (REF ) and (), we reach the following expression
 
I(l'c ,l'T)4GN(5) I(l',l'c ,l'T)L2c T=1(l'c)(l'T)-1+F1(l'T)2 -3(l'T)3-1 (l'T)4          +[F3-2(l'T)4](l'c )2 -[3+3(l'T)4] (l'c )4,                    1,2,3,F1<0,  1,3, F3>0,   
where \(\tilde{\mathcal {I}}(l^{\prime }\Lambda _c ,l^{\prime }T)\)  is the redefined \(I(l^{\prime },l^{\prime }\Lambda _c ,l^{\prime }T)\)  which is given by eq.(REF ) and \(F_3\)  is a numerical constant given by eq.(REF ) in appendix . It is observed that \(\tilde{\mathcal {I}}(l^{\prime }\Lambda _c ,l^{\prime }T)\)  does not depend on the length of the subsystems \(l\)  which is in complete agreement with the results obtained in [1]}. The term including \(F_3\)  is always positive and therefore the non-conformality in the limit of intermediate temperature, unlike the high energy limit, increases \(\tilde{\mathcal {I}}(l^{\prime }\Lambda _c ,l^{\prime }T)\) . In the limit of \(l^{\prime }\Lambda _c\rightarrow 0\) , we reproduce the results obtained for the AdS black hole [1]}. Since the term including \(F_1\)  is proportional to the area of entangling region the HMI obeys an area low even at intermediate temperature. We want to study the HMI in the limit of \(l^{\prime }\rightarrow 0\)  which corresponds to the case that the two subsystems \(A\)  and \(B\)  touch each other. By taking \(l^{\prime }\rightarrow 0\)  limit from eq.(REF ) we reach the following expression

I(l',l'c ,l'T)l'0=14GN(5)-1(Ll')2+F1(LT)2+F3(Lc )2,   1,F1<0, F3>0.

The first term in the above equation is the leading term which obeys an area low divergence in the limit \(l^{\prime }\rightarrow 0\) . The second and third terms are finite sub-leading terms which scale with the area of the strip region \(L^2\)  times to the \(T^2\)  and \(\Lambda _c^2\) , respectively. In the subsection REF  we observe that at the high temperature there is a thermal entropy contribution to the entanglement entropy which scales with the volume of the strip region \(lL^2\) . The thermal entropy contribution is not appear in eq.(REF ) and hence this equation only measure pure quantum entanglement [3]}, [4]}, [1]}.
